🧮 在线计算器：VDI 2230 Bolted Joint Calculation — The full 14-step calculation chain (R0–R13), with six strength checks.
Six strength checks and the tightening torque: R7–R13
R1–R6 answered “how much preload to apply”; R7–R13 answer “can the bolt take it”.
1. R7 — assembly stress check (§5.5.1)
This is the first gate: the bolt does not exceed yield strength during assembly.
The standard allows the use of a fraction of the yield strength (usually $\nu = 0.9$, i.e. 90%); the allowable assembly comparison stress is (VDI 2230:2015, §5.5.1, Eq. R7/1):
$$ \sigma_{red,Mzul} = \nu \cdot R_{p0.2\min} \tag{R7/1} $$The allowable assembly preload $F_{Mzul}$ accounts for the combined stress of tension and torsion that the bolt carries at the same time during tightening (VDI 2230:2015, §5.5.1, Eq. R7/2):
$$ F_{Mzul} = A_0 \cdot \frac{\nu \cdot R_{p0.2\min}}{\sqrt{1 + 3 \left[\frac{3}{2} \frac{d_2}{d_0}\left(\frac{P}{\pi \cdot d_2} + 1.155 \mu_{G\min}\right)\right]^2}} \tag{R7/2} $$Check condition (Eq. R7/3):
$$ F_{Mzul} \geq F_{M\max} \tag{R7/3} $$If it is not met, the standard requires explicitly (VDI 2230:2015, §5.5.1, p.23):
“If the requirement is not met, a larger bolt nominal diameter is to be selected and the calculation repeated from R2.”
2. R8 — service stress check (§5.5.2)
The total bolt force under service contains the preload and the additional bolt force (VDI 2230:2015, §5.5.2, Eq. R8/1):
$$ F_{S\max} = F_{Mzul} + \Phi_{en}^{*} \cdot F_{A\max} - \Delta F_{Vth} \tag{R8/1} $$[!IMPORTANT] Sign convention for the thermal effect Note: here, if $\Delta F_{Vth} > 0$ (the thermal effect reduces the preload), set $\Delta F_{Vth} = 0$ — this is the conservative direction, making sure the check is based on the maximum possible bolt force (VDI 2230:2015, R8, p.24).
The maximum tensile stress and torsional stress (Eq. R8/2, R8/3):
$$ \sigma_{z\max} = F_{S\max} / A_0 \tag{R8/2} $$$$ \tau_{\max} = M_G / W_P \tag{R8/3} $$The standard recommends a reduction factor $k_\tau = 0.5$ to account for the decay of torsional stress during operation — because embedding and vibration partly release the torque introduced at assembly (VDI 2230:2015, §5.5.2, Eq. R8/4):
$$ \sigma_{red,B} = \sqrt{\sigma_{z\max}^2 + 3(k_\tau \cdot \tau_{\max})^2} \tag{R8/4} $$Check condition and safety factor (Eq. R8/5):
$$ S_F = \frac{R_{p0.2\min}}{\sigma_{red,B}} \geq 1.0 \tag{R8/5-2} $$3. R9 — fatigue stress check (§5.5.3)
If the joint carries a dynamic load (the external force alternates between $F_{A\max}$ and $F_{A\min}$), fatigue must be checked.
The stress amplitude (VDI 2230:2015, §5.5.3, Eq. R9/1):
$$ \sigma_a = \frac{F_{SAo} - F_{SAu}}{2 A_S} \tag{R9/1} $$where $F_{SAo}$ and $F_{SAu}$ are the maximum and minimum additional bolt forces.
The standard gives empirical formulas for the bolt fatigue limit ($N_D \geq 2 \times 10^6$):
Rolled after heat treatment (SV) (VDI 2230:2015, Eq. R9/5-1):
$$ \sigma_{ASV} = 0.85 \cdot (150/d + 45) \tag{R9/5-1} $$Rolled before heat treatment (SG) (Eq. R9/5-2):
$$ \sigma_{ASG} = (2 - F_{Sm}/F_{0.2\min}) \cdot \sigma_{ASV} \tag{R9/5-2} $$Check condition (Eq. R9/3, R9/4):
$$ \sigma_a \leq \sigma_{AS} \tag{R9/3} $$$$ S_D = \frac{\sigma_{AS}}{\sigma_a} \geq 1.0 \quad (\text{recommended} \geq 1.2) \tag{R9/4} $$4. R10 — bearing pressure check (§5.5.4)
The bearing pressure under the bolt head / nut must not exceed the limiting surface pressure $p_G$ of the clamped-part material, otherwise it causes creep and preload loss.
The standard explains (VDI 2230:2015, §5.5.4, p.26):
“Surface pressures which cause creep (time-dependent plastic flowing) in conjunction with a loss of preload should not become effective.”
Assembly state (Eq. R10/1):
$$ p_{M\max} = F_{Mzul} / A_{p\min} \leq p_G \tag{R10/1} $$Service state (Eq. R10/2):
$$ p_{B\max} = (F_{V\max} + F_{SA\max} - \Delta F_{Vth}) / A_{p\min} \leq p_G \tag{R10/2} $$Safety factor (Eq. R10/4):
$$ S_p = p_G / p_{M/B\max} \geq 1.0 \tag{R10/4} $$5. R11 — minimum engagement depth (§5.5.5)
For a tapped-thread joint (ESV), the thread engagement depth must be sufficient to avoid stripping the internal thread (VDI 2230:2015, §5.5.5, Eq. R11/1):
$$ F_{mS} \leq \min(F_{mGM},\; F_{MGS}) \tag{R11/1} $$That is, the maximum breaking force of the bolt must be less than the stripping force of the internal thread or the bolt thread.
In Bild 36 the standard gives a chart of the minimum relative engagement depth $m_{eff}/d$ for standard threads M4–M39; typically:
- Steel on steel: $m_{eff} \approx 0.8d \sim 1.0d$
- Steel on cast iron / aluminium: $m_{eff} \approx 1.5d \sim 2.5d$
6. R12 — slip safety (§5.5.6)
The transverse force must be transmitted through the friction at the interface. The minimum residual clamping force (VDI 2230:2015, §5.5.6, Eq. R12/1):
$$ F_{KR\min} = \frac{F_{Mzul}}{\alpha_A} - (1 - \Phi_{en}^{*}) F_{A\max} - F_Z - \Delta F_{Vth} \tag{R12/1} $$Slip safety factor (Eq. R12/4):
$$ S_G = \frac{F_{KR\min}}{F_{KQerf}} > 1.0 \tag{R12/4} $$Recommended values from the standard (VDI 2230:2015, §5.5.6, p.28):
- Static load: $S_G \geq 1.2$
- Alternating transverse force: $S_G \geq 1.8$
If $S_G$ is not met, the standard also provides a shear check as a backup — checking whether the bolt shank would be sheared off at the interface (Eq. R12/5–R12/7).
7. R13 — tightening torque (§5.4.3)
The final step turns the design result into installation parameters the workshop can carry out (VDI 2230:2015, §5.4.3, Eq. R13/1):
$$ M_A = F_{Mzul} \left[0.16 \cdot P + 0.58 \cdot d_2 \cdot \mu_{G\min} + \frac{D_{Km}}{2} \cdot \mu_{K\min}\right] \tag{R13/1} $$The three terms represent, respectively:
| Term | Physical meaning |
|---|---|
| $0.16 \cdot P$ | helix-angle effect (pitch contribution) |
| $0.58 \cdot d_2 \cdot \mu_{G\min}$ | thread-flank friction torque |
| $\frac{D_{Km}}{2} \cdot \mu_{K\min}$ | bolt-head / nut bearing-surface friction torque |
[!NOTE] The make-up of the torque In a typical case, about 50% of the total tightening torque is spent on the bearing-surface friction, about 40% on the thread friction, and only about 10% is turned into effective preload. This is why the friction coefficient is so critical to preload design.
8. The logical hierarchy of the six checks
R7 assembly stress check → bolt does not exceed yield at assembly (static, one-off)
R8 service stress check → does not exceed yield in service (static, continuous)
R9 fatigue stress check → no fatigue fracture under alternating load (dynamic, long-term)
R10 bearing pressure check → no crushing under the head / nut (material, creep)
R11 engagement depth check → internal thread not stripped (geometry, ESV only)
R12 slip safety check → interface does not slip (friction, transverse force)
R13 tightening torque → turned into workshop parameters (output)
If R7 is not met → select a larger bolt and recompute from R2 If R8–R12 are not met → adjust bolt size / property class / tightening method / friction coefficient / external force
Data basis and accuracy statement
All formulas in this article are from VDI 2230 Blatt 1:2015-11, §4.2 calculation summary and §5.5. The recommended safety-factor values are from the respective sections of the standard.
Disclaimer: This article is for engineering teaching reference only.
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